Given two positive numbers, a (the dividend) and n (the divisor), amodulon (abbreviated as a mod n) is the remainder of the Euclidean division of a by n. For instance, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 leaves a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 has a quotient of 3 and leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Note that doing the division with a calculator will not show the result referred to here by this operation; the quotient will be expressed as a decimal fraction.)

Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of n is 0 to n − 1. (n mod 1 is always 0; n mod 0 is undefined, possibly resulting in a "Division by zero" error in computer programming languages) See modular arithmetic for an older and related convention applied in number theory.

When either a or n is negative, the naive definition breaks down and programming languages differ in how these values are defined.

In mathematics the result of the modulo operation is the remainder of the Euclidean division. However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the programming language and/or the underlying hardware.

In nearly all computing systems, the quotientq and the remainder r satisfy

This means that, if the remainder is nonzero, there are two possible choices for the remainder, one negative and the other positive, and there are also two possible choices for the quotient. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a and n.[5] Pascal and Algol68 give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C89, do not define a result if either of n or a is negative. See the table for details. a modulo 0 is undefined in the majority of systems, although some do define it to be a.

Many implementations use truncated division where the quotient is defined by truncationq = trunc(a/n), in other words it is the first integer in the direction of 0 from the exact rational quotient, and the remainder by r=a − nq. Informally speaking the quotient is "rounded towards zero", and the remainder therefore has the same sign as the dividend.

Knuth[5] described floored division where the quotient is defined by the floor functionq=floor(a/n) and the remainder r is

Here the quotient is always rounded downwards (even if it is already negative) and the remainder has the same sign as the divisor.

Raymond T. Boute[6] describes the Euclidean definition, which is the one in which the remainder is always positive or 0, and is therefore consistent with the division algorithm (see Euclidean division). This definition is marked as "Always positive" in the table. Let q be the integer quotient of a and n, then:

Boute argues that Euclidean division is superior to the other ones in terms of regularity and useful mathematical properties, although floored division, promoted by Knuth, is also a good definition. Despite its widespread use, truncated division is shown to be inferior to the other definitions.

Common Lisp also defines round- and ceiling-division where the quotient is given by q=round(a/n), q=ceil(a/n). IEEE 754 defines a remainder function where the quotient is a/n rounded according to the round to nearest convention.

Some calculators have a mod() function button, and many programming languages have a mod() function or similar, expressed as mod(a, n), for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as

a % n

or

a mod n

or equivalent, for environments lacking a mod() function (note that 'int' inherently produces the floor value of a/n)

Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, there are faster alternatives on some hardware. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation:

x % 2n == x & (2n - 1).

Examples (assuming x is a positive integer):

x % 2 == x & 1

x % 4 == x & 3

x % 8 == x & 7.

In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.[8]

Optimizingcompilers may recognize expressions of the form expression % constant where constant is a power of two and automatically implement them as expression & (constant-1). This can allow the programmer to write clearer code without compromising performance. (Note: This will not work for the languages whose modulo have the sign of the dividend (including C), because if the dividend is negative, the modulo will be negative; however, expression & (constant-1) will always produce a positive result. So special treatment has to be made when the dividend can be negative.)

^"ISO/IEC 14882:2003 : Programming languages -- C++". 5.6.4: ISO, IEC. 2003.. "the binary % operator yields the remainder from the division of the first expression by the second. .... If both operands are nonnegative then the remainder is nonnegative; if not, the sign of the remainder is implementation-defined".